Who was Fibonacci?
The "greatest European mathematician of the middle ages", his full
name was Leonardo of Pisa, or Leonardo Pisano in Italian since he was
born in Pisa (Italy), the city with the famous Leaning Tower, about 1175 AD.
Pisa was an important commercial town in its day and had links with many
Mediterranean ports. Leonardo's father, Guglielmo Bonacci, was a kind of
customs officer in the North African town of Bugia now called Bougie where
wax candles were exported to France. They are still called "bougies" in
French, but the town is a ruin today says D E Smith (see below).
So Leonardo grew up with a North African education under the Moors and
later travelled extensively around the Mediterranean coast. He would have
met with many merchants and learned of their systems of doing arithmetic.
He soon realised the many advantages of the "Hindu-Arabic" system over
all the others.
D E Smith points out that another famous Italian - St Francis of
Assisi (a nearby Italian town) - was also alive at the same time as
Fibonacci: St Francis was born about 1182 (after Fibonacci's around 1175)
and died in 1226 (before Fibonacci's death commonly assumed to be
around 1250).
By the way, don't confuse Leonardo of Pisa with Leonardo da Vinci!
Vinci was just a few miles from Pisa on the way to Florence, but Leonardo
da Vinci was born in Vinci in 1452, about 200 years after the death of
Leonardo of Pisa (Fibonacci).
His names
Fibonacci
Leonardo of Pisa is now known as Fibonacci [pronounced fib-on-arch-ee]
short for filius Bonacci.
There are a couple of explanations for the meaning of Fibonacci:
1. Fibonacci is a shortening of the Latin "filius Bonacci", used in the title of his book
Libar Abaci (of which mmore later), which means "the son of Bonaccio". His
father's name was Guglielmo Bonaccio. Fi'-Bonacci is like the English names of
Robin-son and John-son. But (in Italian) Bonacci is also the plural of Bonaccio;
therefore, two early writers on Fibonacci (Boncompagni and Milanesi) regard
Bonacci as his family name (as in "the Smiths" for the family of John Smith).
Fibonacci himself wrote both "Bonacci" and "Bonaccii" as well as "Bonacij"; the
uncertainty in the spelling is partly to be ascribed to this mixture of spoken Italian
and written Latin, common at that time. However he did not use the word
"Fibonacci". This seems to have been a nickname probably originating in the
works of Guillaume Libri in 1838, accordigng to L E Sigler's in his Introduction to
Leonardo Pisano's Book of Squares (see Fibonacci's Mathematical Books
below).
2. Others think Bonacci may be a kind of nick-name meaning "lucky son" (literally,
"son of good fortune").
Statue of Fibonacci
Other names
Other names
He is perhaps more correctly called Leonardo of Pisa or, using a latinisation of his
He
name,
is perhaps
Leonardo
more
Pisano.
correctly
Occasionally
called Leonardo
he alsoofwrote
Pisa Leonardo
or, using aBigollo
latinisation
since,
of in
his
Tuscany,
name, bigollo
Leonardo
means
Pisano.
a traveller.
Occasionally he also wrote Leonardo Bigollo
since,
We shall
in Tuscany,
just call him
bigollo
Fibonacci
meansas
a traveller.
do most modern authors, but if you are looking him
We
up in
shall
older
justbooks,
call him
be Fibonacci
prepared to
assee
do most
any ofmodern
the above
authors,
variations
but if of
you
hisare
name.
looking
[With thanks
him uptoinProf.
olderClaudio
books,Giomini
be prepared
of Rome
to see
for any
helpofonthe
theabove
Latin variations
and Italianofnames
his
in this
name.
section.]
[With thanks to Prof. Claudio Giomini of Rome for help on the Latin and Italian
names in this section.]
Fibonacci's Mathematical Contributions
Fibonacci's Mathematical Contributions
Introducing the Decimal Number system into Europe
Introducing the Decimal Number system into Europe
He was one of the first people to introduce the Hindu-Arabic number system into
Europe - the positional system we use today - based on ten digits with its decimal point
He
andwas
a symbol
one of for
thezero:
first people to introduce the Hindu-Arabic number system into
Europe
1 2 3 4 -5the
6 7positional
890
system we use today - based on ten digits with its decimal
point
His book
and on
a symbol
how tofor
dozero:
arithmetic in the decimal system, called Liber abbaci (meaning
1Book
2 3 4of5the
6 7Abacus
8 9 0 or Book of Calculating) completed in 1202 persuaded many
His
European
book onmathematicians
how to do arithmetic
of his day
in the
todecimal
use thissystem,
"new" system.
called Liber abbaci
(meaning
The book Book
describes
of the(in
Abacus
Latin) the
or Book
rulesofwe
Calculating)
all now learn
completed
at elementary
in 1202
school for
persuaded
adding numbers,
many European
subtracting,
mathematicians
multiplying andofdividing,
his day to
together
use thiswith
"new"
many
system.
problems to
The
illustrate
book the
describes
methods:
(in Latin) the rules we all now learn at elementary school for
adding
174
numbers,
+ 1 7 subtracting,
41 7 4 xmultiplying
1 7 4 ÷ and
28 dividing, together with many
problems
2 8 to illustrate
2 8 the methods:
28
is
1----7 4 + ----1 7 4 -------1 7 4 x 1 7 4 ÷ 28
2 20 82
1 24 86
34
2 8 0 + 6 isremainder 6
--------- ---------- -------1 3 9 2
202
146
3 ------4 8 0 + 6 remainder 6
--------14 38 97 22
------Let's first of all look at 4the
8 7Roman
2
number system still in use in Europe at that time
(1200) and see how awkward
------- it was for arithmetic.
Let's first of all look at the Roman number system still in use in Europe at that
time
Roman
(1200)
Numerals
and see how awkward it was for arithmetic.
The Numerals are letters
Roman Numerals
The Numerals are letters
The method in use in Europe until then used the Roman numerals:
I = 1,
The
V =method
5,
in use in Europe until then used the Roman numerals:
X
I ==1,
10,
V = 5,
X = 10,
LL==50,
50,
C
=
100,
C = 100,
D == 500
500 and
and
D
M
=
1000
M = 1000
You can
can still
still see
see them
them used
used on
on foundation
foundation stones
stones of
of old
old buildings
buildings and
and on
on some
some
You
clocks.
clocks.
The Additive
Additive rule
rule
The
The
simplest
system
would be
be merely
merely to
to use
use the
the letters
letters for
for the
the values
values as
as in
in the
the
The simplest system would
table above,
above, and
and add
add the
the values
values for
for each
each letter
letter used.
used.
table
For
instance,
13
could
be
written
as
XIII
or
perhaps
IIIX or
or even
even IIXI.
IIXI. This
This occurs
occurs in
For instance, 13 could be written as XIII or perhaps IIIX
the
Roman
language
ofof
Latin
where
2323
is is
spoken
asas
tres
etet
viginti
which
translates
in
the
Roman
language
Latin
where
spoken
tres
viginti
which
as
three
and
twenty.
You
may
remember
the
nursery
rhyme
Sing
a
Song
ofa
translates as three and twenty. You may remember the nursery rhyme Sing
Sixpence
which
begins
Song of Sixpence which begins
Sing aa song
song of
of sixpence
sixpence
Sing
A
pocket
full
of
rye
A pocket full of rye
Four and
and twenty
twenty blackbirds
blackbirds
Four
Baked
in
a
pie...
Baked in a pie...
Above 100,
100, the
the Latin
Latin words
words use
use the
the same
same order
order as
as we
we do
do in
in English,
English, so
so that
that
Above
whereas
35
is
quinque
et
triginta
(5
and
30),
235
is
ducenti
triginta
quinque
(two
whereas 35 is quinque et triginta (5 and 30), 235 is ducenti triginta quinque (two
hundred thirty
thirty five).
five).
hundred
In
this
simple
system,
using addition
addition only,
only, 99
99 would
would be
be 90+9
90+9 or,
or, using
using only
only the
the
In this simple system, using
numbers
above,
50+10+10+10
+
5+1+1+1+1
which
translates
to
LXXXXVIIII
and
numbers above, 50+10+10+10 + 5+1+1+1+1 which translates to LXXXXVIIII and
by the
the same
same method
method 1998
1998 would
would be
be written
written by
by the
the Romans
Romans as
as
by
MDCCCCLXXXXVIII.
But
some
numbers
are
long
and
it
is
this
where, ifif we
we
MDCCCCLXXXXVIII. But some numbers are long and it is this isis where,
agree to
to let
let the
the order
order of
of letters
letters matter
matter we
we can
can also
also use
use subtraction.
subtraction.
agree
The
subtractive
rule
The subtractive rule
The Roman
Roman language
language (Latin)
(Latin) also
also uses
uses aa subtraction
subtraction principle
principle so
so that
that whereas
whereas 20
20 is
The
viginti
19
is
"1
from
20"
or
undeviginti.
We
have
it
in
English
when
we
say
the
time
is viginti 19 is "1 from 20" or undeviginti. We have it in English when we say the
is "10
7"towhich
is notisthe
as "7as10".
The The
first means
10 minutes
before ( or
time
is to
"10
7" which
notsame
the same
"7 10".
first means
10 minutes
subtracted
from)
7
0'clock,
whereas
the
second
means
10
minutes
added
to (or
before ( or subtracted from) 7 0'clock, whereas the second means 10 minutes
after) 7too'clock.
This
is also reflected
in Roman
numerals.
abbreviation
added
(or after)
7 o'clock.
This is also
reflected
in RomanThis
numerals.
This makes
the
order
of
letters
important.
So
if
a
smaller
value
came
before
the
next
larger one,
abbreviation makes the order of letters important. So if a smaller value came
it was subtracted
and one,
if it came
it was added.
before
the next larger
it wasafter,
subtracted
and if it came after, it was added.
For
example,
XI
means
10+1=11
(since
the smaller
smaller one
one comes
comes after
after the
the larger
larger ten)
For example, XI means 10+1=11 (since the
but IX
1 less
thanthan
10 or109.or 9.
ten)
butmeans
IX means
1 less
But
8
is
still
written
as
VIII
(not IIX).
IIX). The
The subtraction
subtraction in
in numbers
numbers was
was only
only of
of aa unit
unit
But 8 is still written as VIII (not
(1,
10
or
100)
taken
away
from
5
of
those
units
(5,
50
or
500
or
from
the
next
larger
(1, 10 or 100) taken away from 5 of those units (5, 50 or 500 or from the next
multiple
of 10 (10,
1000).
larger
multiple
of 10100
(10,or100
or 1000).
Using
this
method,
1998
would
be written
written much
much more
more compactly
compactly as
as MCMXCVIII
MCMXCVIII but
Using this method, 1998 would be
thisthis
takes
a little
more
timetime
to interpret:
1000
+ (100
lessless
than
1000)
+ (10
less than
but
takes
a little
more
to interpret:
1000
+ (100
than
1000)
+ (10
100)
+
5
+
1
+
1
+
1.
Note
that
in
the
UK
we
use
a
similar
system
for
time
less than 100) + 5 + 1 + 1 + 1. Note that in the UK we use a similar systemwhen
for
6:50
is
often
said
as
"ten
to
7"
as
well
as
"6
fifty",
similarly
for
"a
quarter
to
4"
time when 6:50 is often said as "ten to 7" as well as "6 fifty", similarly for "a
meaning
In the USA,
sometimes
as "10
of 7". as "10 of 7".
quarter
to 3:45.
4" meaning
3:45. 6:50
In theisUSA,
6:50 isspoken
sometimes
spoken
Look out
out for
for Roman
Roman numerals
numerals used
used as
as the
the date
date aa film
film was
was made,
made, often
often recorded
recorded on
Look
the
screen
which
gives
its
censor
certification
or
perhaps
the
very
last
image
on the screen which gives its censor certification or perhaps the very last image
of the
the movie
movie giving
giving credits
credits or
or copyright
copyright information.
information.
of
Arithmetic
with
Arithmetic with Roman
Roman Numerals
Numerals
Arithmetic was
was not
not easy
easy in
in the
the Roman
Roman system:
system:
Arithmetic
CLXXIIII added
added to
to XXVIII
XXVIII isis CCII
CCII
CLXXIIII
CLXXIIII
less
XXVIII
is
CXXXXVI
CLXXIIII less XXVIII is CXXXXVI
For more
more on
on Roman
Roman Numerals,
Numerals, see
see the
the excellent
excellent Frequently
Frequently Asked
Asked Questions
Questions
For
on
Roman
Numerals
at
Math
Forum.
on Roman Numerals at Math Forum.
Stems from Fibonacci Decimal Positioning
Stems from Fibonacci Decimal Positioning
The Decimal
Decimal Positional
Positional System
System
The
The system
system that
that Fibonacci
Fibonacci introduced
introduced into
into Europe
Europe came
came from
from India
India and
and Arabia
Arabia
The
and
used
the
Arabic
symbols
1,
2,
3,
4,
5,
6,
7,
8,
9
with,
most
importantly,
and used the Arabic symbols 1, 2, 3, 4, 5, 6, 7, 8, 9 with, most importantly, aa
symbol for
for zero
zero 0.
0.
symbol
With Roman
Roman numbers,
numbers, 2003
2003 could
could be
be written
writtenas
as MMIII
MMIII or,
or, just
just as
as clearly,
clearly, itit could
could
With
be
written
as
IIIMM
the
order
does
not
matter
since
the
values
of
the
letters
be written as IIIMM - the order does not matter since the values of the letters are
are added
to make
the number
the original
(unabbreviated)
system.
added
to make
the number
in theinoriginal
(unabbreviated)
system.
WithWith
the the
abbreviated
system
of
IX
meaning
9,
then
the
order
did
matter
but
it
seems
this
abbreviated system of IX meaning 9, then the order did matter but it seems this
sytem was
was not
not often
often used
used in
in Roman
Roman times.
times.
sytem
In
the
"new
system",
the
order
does
matter
always since
since 23
23 is
is quite
quite aa different
different
In the "new system", the order does matter always
number to
to 32.
32. Also,
Also, since
since the
the position
position of
of each
each digit
digit isis important,
important, then
then we
we may
may
number
need
a
zero
to
get
the
digits
into
their
correct
places
(columns)
eg
2003
which
need a zero to get the digits into their correct places (columns) eg 2003 which
has no
no tens
tens and
and no
no hundreds.
hundreds. (The
(The Roman
Roman system
system would
would have
have just
just omitted
omitted the
the
has
values not
not used
used so
so had
had no
no need
need of
of "zero".)
"zero".)
values
This
decimal
positional
system,
as
we
call it,
it, uses
uses the
the ten
ten symbols
symbols of
of Arabic
Arabic
This decimal positional system, as we call
origin and
and the
the "methods"
"methods" used
used by
by Indian
Indian Hindu
Hindu mathematicians
mathematicians many
many years
years
origin
before
they
were
imported
into
Europe.
It
has
been
commented
that
in
India, the
the
before they were imported into Europe. It has been commented that in India,
concept of
of nothing
nothing isis important
important in
in its
its early
early religion
religion and
and philosophy
philosophy and
and so
so itit was
was
concept
much
more
natural
to
have
a
symbol
for
it
than
for
the
Latin
(Roman)
and
Greek
much more natural to have a symbol for it than for the Latin (Roman) and Greek
systems.
systems.
"Algorithm"
"Algorithm"
Earlier the
the Persian
Persian author
author Abu
Abu ‘Abd
‘Abd Allah,
Allah, Mohammed
Mohammed ibn
ibn Musa
Musa al-Khwarizmi
al-Khwarizmi
Earlier
(usually
abbreviated
to
Al-Khwarizmi
had
written
a
book
which
included
the rules
(usually abbreviated to Al-Khwarizmi had written a book which included the
rules
of arithmetic for the decimal number system we now use, called Kitab al jabr wa‘lmuqabala
of arithmetic
(Rules
forofthe
restoring
decimaland
number
equating)
system
dating
we from
now use,
aboutcalled
825 AD.
Kitab
D al
E jabr
Knuth (in
thewa‘l-muqabala
errata for the second
(Rules edition
of restoring
and third
and equating)
edition of dating
his "Fundamental
from about 825
Algorithms")
AD. D E
gives
Knuth
the (in
fullthe
name
errata
above
for the
andsecond
says it can
edition
be translated
and third edition
as Father
of his
of "Fundamental
Abdullah,
Mohammed,
Algorithms")
songives
of Moses,
the fullnative
nameof
above
Khwarizm.
and says
He itwas
canan
beastromomer
translated as
to Father
the caliph
of at
Baghdad
Abdullah,
(now
Mohammed,
in Iraq). son of Moses, native of Khwarizm. He was an
astromomer to the caliph at Baghdad (now in Iraq).
Al-Khowârizmî is the region south and to the east of the
Al-Khowârizmî
Aral Sea around
is the region
the town
south
nowand
called
to the
Khiva
east(or
Urgench)
of the Aral
on the
SeaAmu
around
Darya
theriver.
townItnow
wascalled
part ofKhiva
the Silk
Route,
(or Urgench)
a majorontrading
the Amu
pathway
Darya between
river. It was
the part
Eastofand
the Silk
Europe.
Route,
Ina1200
major
it was
trading
in Persia
pathway
butbetween
today is in
the
Uzbekistan,
East and part
Europe.
of the
In former
1200 it USSR,
was in north
Persiaofbut
Iran,
today is in Uzbekistan,
which gainedpart
its of
independence
the former USSR,
in 1991.
north of Iran, which gained its independence in
1991.
Prof Don Knuth has a picture of a postage stamp issued by the USSR in 1983 to
commemorate al-Khowârizmî 1200 year anniversary of his probable birth date.
From
Profthe
Don
titleKnuth
of thishas
book
a picture
Kitab alofjabr
a postage
w'al-muqabala
stamp issued
we derive
by the
our
USSR
modern
in 1983
wordto
algebra.
commemorate al-Khowârizmî 1200 year anniversary of his probable birth date.
TheFrom
Persian
the title
author's
of this
name
bookisKitab
commemorated
al jabr w'al-muqabala
in the wordwe
algorithm.
derive our
It modern
has changed
over
word
the algebra.
years from an original European pronunciation and latinisation of algorism.
Algorithms
The Persian
were author's
known ofname
before
is commemorated
Al-Khowârizmî's in
writings,
the word
(foralgorithm.
example, It
Euclid's
has
Elements
changed
is over
full ofthe
algorithms
years from
for an
geometry,
original including
Europeanone
pronunciation
to find the greatest
and latinisation
common
divisor
of algorism.
of two numbers
Algorithms
called
were
Euclid's
knownalgorithm
of beforetoday).
Al-Khowârizmî's writings, (for
The
example,
USA Library
Euclid's
of Congress
Elements has
is full
a of
listalgorithms
of citationsfor
of geometry,
Al-Khowârizmî
including
and his
oneworks.
to
Ourfind
modern
the greatest
word "algorithm"
common divisor
does not
of two
just numbers
apply to the
called
rules
Euclid's
of arithmetic
algorithm
buttoday).
means
anyThe
precise
USA set
Library
of instructions
of Congress for
hasperforming
a list of citations
a computation
of Al-Khowârizmî
whether
and
thishis
be a
method
works.
followed by humans, for example:
Our modern word "algorithm" does not just apply to the rules of arithmetic but
means any
precise set of instructions for performing a computation whether
a cooking
recipe;
this
be
a
method
a knitting pattern; followed by humans, for example:
travel instructions;
a car
a cooking
manualrecipe;
page for example, on how to remove the gear-box;
a medical
a knitting
procedure
pattern; such as removing your appendix;
a calculation
travel instructions;
by human computors : two examples are:
William
a carShanks
manualwho
page
computed
for example,
the value
on how
of pi
to to
remove
707 decimal
the gear-box;
places by hand last
century
a medical
over about
procedure
20 years
such
upas
to removing
1873 - butyour
he was
appendix;
wrong at the 526-th place when it
wasachecked
calculation
by desk
by human
calculators
computors
in 1944!
: two examples are:
Earlier
William
Johann
Shanks
Dase
who
hadcomputed
computedthe
pi value
correctly
of pitoto205
707decimal
decimalplaces
placesinby
1844
hand
when
aged
last20
century
but thisover
wasabout
done20
completely
years up in
to his
1873
head
- but
just
hewriting
was wrong
the number
at the 526-th
down after
working
place on
when
it forit two
wasmonths!!
checked by desk calculators in 1944!
or mechanically
Earlier Johannby
Dase
machines
had computed
(such aspiplacing
correctly
chips
to 205
anddecimal
components
places
at in
correct
1844
places
whenonaged
a circuit
20 but
board
this to
was
go done
insidecompletely
your TV) in his head just writing the number
down after working on it for two months!!
or mechanically by machines (such as placing chips and components at correct
places on a circuit board to go inside your TV)
or
or automatically
automatically by
by electronic
electronic computers
computers which
which store
store the
the instructions
instructions as
as well
well as
as
data
to
work
on.
data to work on.
See D
D EE Knuth,
Knuth, The
The Art
Art of
of Computer
Computer Programming
Programming Volume
Volume 1:
1: Fundamental
Fundamental
See
Algorithms
(now
in
its
Third
Edition,
1997)pages
1-2.
There
is
an English
English
Algorithms (now in its Third Edition, 1997)pages 1-2. There is an
translation of
of the
the "..
".. al
al jabr
jabr .."
.." book:
book: LL C
C Karpinski
Karpinski Robert
Robert of
of Chester's
Chester's Latin
Latin
translation
Translation
...
of
al-Khowarizmi
published
in
New
York
in
1915.
[Note
the
Translation ... of al-Khowarizmi published in New York in 1915. [Note the
variation in
in the
the spelling
spelling of
of "Al-Khowârizmî"
"Al-Khowârizmî" here
here -- this
this isis not
not unusual!
unusual! Other
Other
variation
spellings
include
al-Khorezmi.]
Ian
Stewart's
The
Problems
of
Mathematics
spellings include al-Khorezmi.] Ian Stewart's The Problems of Mathematics
(Oxford) 1992,
1992, ISBN:
ISBN: 0-19-286148-4
0-19-286148-4 has
has aa chapter
chapter on
on algorithms
algorithms and
and the
the history
history of
(Oxford)
the
name:
chapter
21:
Dixit
Algorizmi.
of the name: chapter 21: Dixit Algorizmi.
TheFibonacci
FibonacciNumbers
Numbers
The
In Fibonacci's
Fibonacci's Liber
Liber Abaci
Abaci book,
book, chapter
chapter 12,
12, he
he introduces
introduces the
the following
following problem
problem
In
(here in
in Sigler's
Sigler's translation
translation -- see
see below):
below):
(here
How
Many
Pairs
of
Rabbits
Are
Created by
by One
One Pair
Pair in
in One
One Year
Year
How Many Pairs of Rabbits Are Created
certain man
man had
had one
one pair
pair of
of rabbits
rabbits together
together in
in aa certain
certain enclosed
enclosed place,
place, and
and one
AA certain
wishes
to
know
how
many
are
created
from
the
pair
in
one
year
when
it
is
the
one wishes to know how many are created from the pair in one year when it is
nature
of them
in ainsingle
month
to bear
another
pair,
andand
in the
second
month
the
nature
of them
a single
month
to bear
another
pair,
in the
second
those born
beartoalso.
month
thosetoborn
bear also.
He
then
goes
on
to
solve
and explain
explain the
the solution:
solution:
He then goes on to solve and
DidFibonacci
Fibonacciinvent
inventthis
thisSeries?
Series?
Did
Fibonacci says
says his
his book
book Liber
Liber Abaci
Abaci (the
(the first
first edition
edition was
was dated
dated 1202)
1202) that
that he
he had
had
Fibonacci
studied the
the "nine
"nine Indian
Indian figures"
figures" and
and their
their arithmetic
arithmetic as
as used
used in
in various
various countries
countries
studied
around
the
Mediterranean
and
wrote
about
them
to
make
their
use
more
around the Mediterranean and wrote about them to make their use more commonly
understoodunderstood
in his native
So heItaly.
probably
included
"rabbit the
problem"
commonly
in Italy.
his native
So hemerely
probably
merelythe
included
from
one
of
his
contacts
and
did
not
invent
either
the
problem
or
the
series
of or
"rabbit problem" from one of his contacts and did not invent either the problem
numbers
which
now
bear
his
name.
the series of numbers which now bear his name.
D EE Knuth
Knuth adds
adds the
the following
following in
in his
his monumental
monumental work
work The
The Art
Art of
of Computer
Computer
D
Programming:
Volume
1:
Fundamental
Algorithms
errata
to
second
edition:
Programming: Volume 1: Fundamental Algorithms errata to second edition:
Before Fibonacci
Fibonacci wrote
wrote his
his work,
work, the
the sequence
sequence F(n)
F(n) had
had already
already been
been discussed
discussed by
Before
Indian
scholars,
who
had
long
been
interested
in
rhythmic
patterns
that
areare
formed
by Indian scholars, who had long been interested in rhythmic patterns that
from one-beat
and two-beat
notes. The
number
of suchof
rhythms
having having
n beatsn
formed
from one-beat
and two-beat
notes.
The number
such rhythms
altogether
is
F(n+1);
therefore
both
Gospala
(before
1135)
and
Hemachandra
(c.
beats altogether is F(n+1); therefore both Gospala (before 1135) and
1150) mentioned
the numbers
1, 2,the
3, 5,
8, 13, 21,
... 3,
explicitly.
Hemachandra
(c. 1150)
mentioned
numbers
1, 2,
5, 8, 13, 21, ... explicitly.
Knuth
refers
to
an
article
by
P
Singh
in
Historia
Mathematica
vol 12
12 (1985)
(1985) pages
Knuth refers to an article by P Singh in Historia Mathematica vol
229-244.
pages 229-244.
Namingthe
theSeries
Series
Naming
was the
the French
French mathematician
mathematician Edouard
Edouard Lucas
Lucas (1842-1891)
(1842-1891) who
who gave
gave the
the name
ItIt was
Fibonacci
numbers
to
this
series
and
found
many
other
important
applications
as
name Fibonacci numbers to this series and found many other important
well as having
of numbers
thatofare
closelythat
related
the Fibonacci
applications
as the
wellseries
as having
the series
numbers
are to
closely
related to
numbers
the
Lucas
Numbers:
2,
1,
3,
4,
7,
11,
18,
29,
47,
...
the Fibonacci numbers - the Lucas Numbers: 2, 1, 3, 4, 7, 11, 18, 29, 47, ...
named after him.
named after him.
Fibonacci memorials to see in Pisa
Fibonacci memorials to see in Pisa
He died in the 1240's and there is now a statue commemorating him located at
the Leaning Tower end of the cemetery next to the Cathedral in Pisa. [With
He died
in theto1240's
andFarhi,
there is
a statue
commemorating
himforlocated
special
thanks
Nicholas
annow
ex-pupil
of Winchester
College,
the at
the Leaning
picture
of the Tower
statue.]end of the cemetery next to the Cathedral in Pisa. [With
special thanks to Nicholas Farhi, an ex-pupil of Winchester College, for the
picture of the statue.]
References
References
D E Smith's History of Mathematics Volume 1, (Dover, 1958 - a reprint of the
orignal version from 1923) gives a complete list of other books that he wrote and
Smith's
History
Mathematics
1, (Dover, 1958 - a reprint of the
is D
aE
fuller
reference
onofFibonacci's
lifeVolume
and works.
orignalisversion
givesof
a Fibonacci
complete list
of other
books
that he
wrote and
There
anotherfrom
brief1923)
biography
which
is part
of Karen
Hunger
is
a
fuller
reference
on
Fibonacci's
life
and
works.
Pashall's (Virginia University) The art of Algebra from from al-Khwarizmi to Viéte:
briefSelection
biography
Fibonacci
whichtoisread
part more
of Karen
Hunger
A There
Study is
in another
the Natural
ofof
Ideas
if you want
about
the
Pashall's
(Virginia
University)
The
art
of
Algebra
from
from
al-Khwarizmi
to Viéte:
history of mathematics.
A Study
in the Natural
Selection
if you (University
want to read
the in
Eight
Hundred
Years Young
by of
A Ideas
F Horadam
ofmore
New about
England)
history
of
mathematics.
The Australian Mathematics Teacher Vol 31, 1985, pages 123-134, is an
Eight Hundred
Years Young
byFibonacci,
A F Horadam
(University
New as
England)
interesting
and readable
article on
his names
and of
origins
well asin
his
The
Australian
Mathematics
Teacher
Vol
31,
1985,
pages
123-134,
is
an
mathematical works. He refers to and expands upon the following article...
interesting
and readable
article on Pisano
Fibonacci,
namesinand
origins Quarterly
as well asvol
The
Autobiography
of Leonardo
R Ehis
Grimm,
Fibonacci
his
mathematical
works.
He
refers
to
and
expands
upon
the
following
article...
11, 1973, pages 99-104.
The
Autobiography
of
Leonardo
Pisano
R
E
Grimm,
in
Fibonacci
Quarterly
Leonard of Pisa and the New Mathematics of the Middle Ages by J and F
vol 11,
1973, Y
pages
99-104.
Gies,
Thomas
Crowell
publishers, 1969, 127 pages, is another book with much
Leonard
of
Pisa
and
the
New Mathematics
on the background to Fibonacci's
life and work. of the Middle Ages by J and F
Gies, vita
Thomas
Y Crowell
publishers,
127Baldassarre
pages, is another
book withRome,
much
Della
e delle
opere di
Leonardo1969,
Pisano
Boncompagni,
on
the
background
to
Fibonacci's
life
and
work.
1854 is the only complete printed version of Fibonacci's 1228 edition of Liber
Della vita e delle opere di Leonardo Pisano Baldassarre Boncompagni,
Abaci.
Rome,
the onlyarchives
complete
version
of Fibonacci's
1228 edition
of
The the 1854
Math is
Forum's
of printed
the History
of Mathematics
discussion
group
Liber Abaci.
contain
a useful discussion on some of the controversial topics of Fibonacci's
The the
Forum's archives
of the
Mathematics
group
names
andMath
life (February
1999). Use
its History
next>>oflink
to follow thediscussion
thread of the
contain a useful
discussion
on some
of the controversial
topics
of Fibonacci's
discussion
through
its 6 emailed
contributions.
It talks about
the uncertainlty
of his
names
and
life
(February
1999).
Use
its
next>>
link
to
follow
the
thread
of the
birth and death dates and his names. It seems that Fibonacci never
referred
to
discussion
through
its
6
emailed
contributions.
It
talks
about
the
uncertainlty
himself as "Fibonacci" but this was a nick-name given to him by later writers. of
his birth and death dates and his names. It seems that Fibonacci never referred
to himself as "Fibonacci" but this was a nick-name given to him by later writers.